NONCOMPACTNESS AND MAXIMUM MOBILITY OF TYPE III RICCI-FLAT SELF-DUAL NEUTRAL WALKER FOUR-MANIFOLDS

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It is shown that a self-dual neutral Einstein four-manifold of Petrov type III, admitting a two-dimensional null parallel distribution compatible with the orientation, cannot be compact or locally homogeneous, and its maximum possible degree of mobility is 3. Díaz-Ramos, García-Río and Vázquez-Lorenzo found a general coordinate form of such manifolds. The present paper also provides a modified version of that coordinate form, valid in a suitably defined generic case and, in a sense, ‘more canonical’ than the usual formulation. Moreover, the local-isometry types of manifolds as above having the degree of mobility equal to 3 are classified. Further results consist in explicit descriptions, first, of the kernel and image of the Killing operator for any torsionfree surface connection with everywhere-nonzero, skew-symmetric Ricci tensor, and, secondly, of a moduli curve for surface connections with the properties just mentioned that are, in addition, locally homogeneous. Finally, hyperbolic plane geometry is used to exhibit examples of codimension-two foliations on compact manifolds of dimensions greater than 2 admitting a transversal torsion-free connection, the Ricci tensor of which is skew-symmetric and nonzero everywhere. No such connection exists on any closed surface, so that there are no analogous examples in dimension 2.